@article{Bose_De Carufel_Shaikhet_Smid_2017, title={Essential Constraints of Edge-Constrained Proximity Graphs}, volume={21}, url={https://jgaa.info/index.php/jgaa/article/view/paper422}, DOI={10.7155/jgaa.00422}, abstractNote={Given a plane forest $F = (V, E)$ of $|V| = n$ points, we find the minimum set $S \subseteq E$ of edges such that the edge-constrained minimum spanning tree over the set $V$ of vertices and the set $S$ of constraints contains $F$. We present an $O(n \log n )$-time algorithm that solves this problem.
We generalize this to other proximity graphs in the constraint setting, such as the <i>relative neighbourhood graph</i>, <i>Gabriel graph</i>, $\beta$<i>-skeleton</i> and <i>Delaunay triangulation</i>. We present an algorithm that identifies the minimum set $S\subseteq E$ of edges of a given plane graph $I=(V,E)$ such that $I \subseteq CG_\beta(V, S)$ for $1 \leq \beta \leq 2$, where $CG_\beta(V, S)$ is the constraint $\beta$-skeleton over the set $V$ of vertices and the set $S$ of constraints. The running time of our algorithm is $O(n)$, provided that the constrained Delaunay triangulation of $I$ is given.}, number={4}, journal={Journal of Graph Algorithms and Applications}, author={Bose, Prosenjit and De Carufel, Jean-Lou and Shaikhet, Alina and Smid, Michiel}, year={2017}, month={Feb.}, pages={389–415} }